# Additive properties of G-Drazin inverse of linear operators

## TL;DR

This paper explores the additive properties of the G-Drazin inverse for linear operators in Banach spaces, establishing conditions under which the sum of two such operators also has a G-Drazin inverse and providing explicit formulas.

## Contribution

It introduces new polynomial conditions ensuring the sum of G-Drazin invertible operators is also G-Drazin invertible and derives explicit inverse representations.

## Key findings

- Sum of operators has G-Drazin inverse under new polynomial conditions
- Explicit formulas for the G-Drazin inverse of the sum
- Conditions extend previous results on additive properties

## Abstract

In this paper, we investigate additive properties of generalized Drazin inverse for linear operators in Banach spaces. Under new polynomial conditions on generalized Drazin invertible operators a and b, we prove their sum has generalized Drazin inverse and give explicit representations of the generalized inverse $(a+b)^d$.